How can one utilize a variable as a callback function name for add_settings_field

Do not use different callbacks, use the the sixth parameter for add_settings_field() instead. That is an array, and you can pass any data to the callback here.

Example:

foreach( $theOptions as $k => $v ) 
{
    add_settings_field(
        $k,
        $v,
        'my_callback',
        $the_options,
        $the_group,
        array (
            'special' => $k
        )
    );
}

function my_callback( $args )
{
    var_dump( $args['special'] );

    switch ( $args['special'] )
    {
        case 'foo':
            // do something
            break;
        case 'bar':
            // do something else
            break;
        default:
            // default handler 
    }
}